Probability Density Function Control And Shannon Entropy Control For Quantum Systems

Probability is the essential characteristic of quantum systems, so it is very important to control the probability density function (PDF) of quantum systems. The classical PDF control method has achieved great success in many classical systems. The property of quantum systems makes it suitable to be controlled by PDF control methods. If the spatial PDF of quantum states can be well controlled, we may essentially enhance quantum control effects, thus satisfy further control requirements of more complicated practical quantum systems. This thesis combines several classical control methods (including optimal control, nonlinear control, adaptive control and Lyapunov control) with PDF control method, and proposes a new quantum control method based on PDF control strategy. The goal of quantum control is extended from controlling the quantum state vector to the control of the detailed spatial probability distribution of the quantum state. And the PDF control theory is extended from real domain to the complex domain, which may yield a new research area of PDF control theory in the complex domain. Since quantum PDF control calls for a distributed external potential field, we’ve also provided a new constructive controller design method to directly design the control field with time-varying shape. What’s more, based on the quantum PDF control, we have extended the traditional Shannon entropy control into quantum area, and proposed a new method which directly controls the Shannon entropy of quantum systems. This may enhance the quantum control performance from many regards. The main contents are as follows:1. Probability density function control of quantum systems. We modeled the quantum PDF control problem, discussed about the selection of quantum PDF, analyzed the controllability of various wave pattens with different shapes, and studied about how to control a real number (the probability) using a complex equation. Then we discussed about how to maintain the quantum PDF unchanged, how to design the quantum PDF controller from both discrete and continuous perspectives, and proposed a new controller design method for the limited external energy case.2. Constructive controller design method of distributed external potential field. Quantum PDF control belongs to stochastic distribution control, and the goal is to control the spatial distribution of the probability. So if a distributed potential field is adopted, the control effect may be improved. If we want the shape of the potential field to be distributed, we’ll need a potential field whose shape can change with the spatial position. Such field can be achieved by adding several fields to yield interference. But how to directly design the superposition of several fields still remains unsolved. We have proposed a constructive design method for the distributed potential field. By refering to a general reference vector, we mixed the variables of both time and space to achieve the goal, which provides a new constructive method for the potential field with time-varying shape.3. Shannon entropy control of quantum systems. Based on classical PDF control, there is also an intense research field of Shannon entropy control, which has achieved good performance in several classical systems. But how to directly control the Shannon entropy still remains unknown. Quantum Shannon entropy can reveal a great deal of information from the perspective of geometrical changes to the density. It shows interesting features about the bond forming and breaking process that are not apparent from the conventional reaction energy profile. The consistency of the Shannon entropy when applied to outcomes of quantum experiments has been analyzed. Recent research about Shannon entropy control in biological control can similarly be extended to the control of electron correlation energy. What’s more, Shannon entropy can play an important role in quantum sliding-mode control and coherent control. So it may act as a new approach to enhance quantum control performance from many regards. In this thesis we have extended classical Shannon entropy control into quantum-area, established the quantum Shannon entropy control model, analyzed the controllability of discrete Shannon entropy, and provided controller design methods based on discretization. Finally we did simulations on several quantum systems to verify our algorithm.